
SIGMA 2 (2006), 021, 10 pages condmat/0602427
http://dx.doi.org/10.3842/SIGMA.2006.021
On the Degenerate Multiplicity of the sl_{2} Loop Algebra for the 6V Transfer Matrix at Roots of Unity
Tetsuo Deguchi
Department of Physics, Faculty of Science, Ochanomizu University, 211 Ohtsuka, BunkyoKu, Tokyo 1128610, Japan
Received October 31, 2005, in final form February 06, 2006; Published online February 17, 2006
Abstract
We review the main result of condmat/0503564. The
Hamiltonian of the XXZ spin chain and the transfer matrix of the
sixvertex model has the sl_{2} loop algebra symmetry if the q
parameter is given by a root of unity, q_{0}^{2N} = 1, for an
integer N. We discuss the dimensions of the degenerate
eigenspace generated by a regular Bethe state in some sectors,
rigorously as follows:
We show that every regular Bethe ansatz eigenvector in the sectors is
a highest weight vector and derive the highest weight
d_{k}^{±},
which leads to evaluation parameters a_{j}.
If the evaluation parameters are distinct, we obtain the
dimensions of the highest weight representation generated by the
regular Bethe state.
Key words:
loop algebra; the sixvertex model; roots of unity representations of quantum groups; Drinfeld polynomial.
pdf (249 kb)
ps (177 kb)
tex (14 kb)
References
 Alcaraz F.C., Grimm U., Rittenberg V.,
The XXZ Heisenberg chain, conformal invariance and the operator
content of c < 1 systems, Nucl. Phys. B, 1989, V.316,
735768.
 Baxter R.J.,
Eightvertex model in lattice statistics and onedimensional
anisotropic Heisenberg chain. I. Some fundamental eigenvectors,
Ann. Phys., 1973, V.76, 124;
II. Equivalence to a generalized Icetype lattice model,
Ann. Phys., 1973, V.76, 2547; III. Eigenvectors of the
transfer matrix and Hamiltonian, Ann. Phys., 1973, V.76,
4871.
 Baxter R.J.,
Completeness of the Bethe ansatz for the six and eight vertex
models, J. Statist. Phys., 2002, V.108, 148,
condmat/0111188.
 Baxter R.J.,
The six and eightvertex models revisited, J. Statist.
Phys., 2004, V.116, 4366, condmat/0403138.
 Braak D., Andrei N.,
On the spectrum of the XXZchain at roots of unity,
J. Statist. Phys., 2001, V.105, 677709, condmat/0106593.
 Chari V., Pressley A.,
Quantum affine algebras, Comm. Math. Phys., 1991, V.142,
261283.
 Chari V., Pressley A.,
Quantum affine algebras at roots of unity, Represent.
Theory, 1997, V.1, 280328, qalg/9609031.
 Chari V., Pressley A., Weyl modules for classical and
quantum affine algebras, Represent. Theory, 2001, V.5, 191223, math.QA/0004174.
 Deguchi T.,
Construction of some missing eigenvectors of the XYZ spin chain at
the discrete coupling constants and the exponentially large
spectral degeneracy of the transfer matrix, J. Phys. A: Math.
Gen., 2002, V.35, 879895, condmat/0109078.
 Deguchi T.,
The 8V CSOS model and the sl_{2} loop algebra symmetry
of the sixvertex model at roots of unity,
Internat. J. Modern Phys. B, 2002, V.16, 18991905,
condmat/0110121.
 Deguchi T.,
XXZ Bethe states as highest weight vectors
of the sl_{2} loop algebra at roots of unity, condmat/0503564.
 Deguchi T.,
The sixvertex model at roots of unity and some highest weight
representations of the sl_{2} loop algebra, in preparation (to be
submitted to the Proceedings of RAQIS'05, Annecy, France).
 Deguchi T., Fabricius K., McCoy B.M.,
The sl_{2} loop algebra symmetry of the sixvertex model at roots
of unity, J. Statist. Phys., 2001, V.102, 701736,
condmat/9912141.
 Fabricius K., McCoy B.M.,
Bethe's equation is incomplete for the XXZ model at roots of
unity, J. Statist. Phys., 2001, V.103, 647678,
condmat/0009279.
 Fabricius K., McCoy B.M.,
Completing Bethe's equations at roots of unity, J. Statist.
Phys., 2001, V.104, 573587, condmat/0012501.
 Fabricius K., McCoy B.M.,
Evaluation parameters and Bethe roots for the sixvertex model at
roots of unity, Progress in Mathematical Physics, Vol. 23
(MathPhys Odyssey 2001), Editors M. Kashiwara and T. Miwa, Boston, Birkhäuser, 2002, 119144, condmat/0108057.
 Fabricius K., McCoy B.M.,
New developments in the eightvertex model,
J. Statist Phys., 2003, V.111, 323337, condmat/0207177.
Fabricius K., McCoy B.M., Functional equations and fusion matrices
for the eightvertex model, Publ. Res. Inst. Math. Sci.,
2004, V.40, 905932, condmat/0311122.
 Fabricius K., McCoy B.M.,
New developments in the eightvertex model II. Chains of odd
length, condmat/0410113.
 Jimbo M., Private communication, July 2004.
 Kac V., Infinite dimensional Lie algebras, Cambridge,
Cambridge University Press, 1990.
 Korepanov I.G., Hidden symmetries in the 6vertex model
of statistical physics, Zap. Nauchn. Sem. S.Peterburg.
Otdel. Mat. Inst. Steklov. (POMI), 1994, V.215, 163177 (English
transl.: J. Math. Sci. (New York), 1997, V.85, 16611670),
hepth/9410066.
 Korepanov I.G., Vacuum curves of the Loperators related to the sixvertex model,
St. Petersburg Math. J., 1995, V.6, 349364.
 Korepin V.E., Bogoliubov N.M., Izergin A.G.,
Quantum inverse scattering method and correlation functions,
Cambridge, Cambridge University Press, 1993.
 Korff C., McCoy B.M.,
Loop symmetry of integrable vertex models at roots of unity,
Nucl. Phys. B, 2001, V.618, 551569, hepth/0104120.
 Lusztig G., Modular representations and
quantum groups, Contemp. Math., 1989, V.82, 5977.
 Lusztig G., Introduction to quantum groups, Boston,
Birkhäuser, 1993.
 Pasquier V., Saleur H.,
Common structures between finite systems and conformal field
theories through quantum groups, Nucl. Phys. B, 1990, V.330,
523556.
 Takhtajan L., Faddeev L.,
Spectrum and scattering of excitations in the onedimensional
isotropic Heisenberg model, J. Sov. Math., 1984, V.24,
241267.
 Tarasov V.O.,
Cyclic monodromy matrices for the Rmatrix of the sixvertex model
and the chiral Potts model with fixed spin boundary conditions,
in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser.
Math. Phys., Vol. 16, River Edge, NJ, World Sci. Publishing,
1992, 963975.
 Tarasov V.O., On the Bethe vectors
for the XXZ model at roots of unity, math.QA/0306032.

